Download George VOUTSADAKIS CATEGORICAL ABSTRACT ALGEBRAIC

REPORTS ON MATHEMATICAL LOGIC
47 (2012), 125–145
DOI:10.4467/20842589RM.12.006.0687
George VOUTSADAKIS
CATEGORICAL ABSTRACT
ALGEBRAIC LOGIC:
COORDINATIZATION IS ALGEBRAIZATION
A b s t r a c t. The methods of categorical abstract algebraic
logic are employed to show that the classical process of the coordinatization of abstract (affine plane) geometry can be viewed
under the light of the algebraization of logical systems. This link
offers, on the one hand, a new perspective to the coordinatization
of geometry and, on the other, enriches abstract algebraic logic
by bringing under its wings a very well-known geometric process,
not known hitherto to be related or amenable to its methods
and techniques. The algebraization takes the form of a deductive
equivalence between two institutions, one corresponding to affine
plane geometry and the other to Hall ternary rings.
Received 6 June 2011
Keywords: Abstract Geometry, Affine Plane Geometry, Coordinatization, Hall Ternary
Rings, Institutions, Algebraic Institutions, Deductive Equivalence, Categorical Abstract
Algebraic Logic, Interpretations, Equivalent Algebraic Semantics
2010 AMS Subject Classification: Primary: 03G27 Secondary: 18A15, 20N10, 51A25
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GEORGE VOUTSADAKIS
1. Introduction
Around the end of the nineteenth and the beginning of the last century,
several authors considered the process of coordinatization of various axiomatically defined abstract geometries. Among them were the affine plane
geometry, affine plane geometry with the Desargues and Pappus properties,
projective plane geometry etc. A classic introductory book to be consulted
on this topic, which partly inspired our investigations and to which the
reader will be referred for several constructions and results that will be
used in the present paper, is Blumenthal’s “A Modern View of Geometry”
[3]. A more recent reference on this material, that is now available online
in its second edition, is Peter Cameron’s “Projective and Polar Spaces”
[6]. Finally, a general reference pertaining to the topic that may prove of
further help to the interested reader is the handbook [5].
The general process of coordinatization consists, generally speaking, of
using the points and incidence structure of a given abstract geometry as
a basis for constructing an abstract algebraic structure. Ordinarily, the
points of the geometry serve as the building blocks for the universe of the
algebraic structure. Then, the structural properties of the geometry are
used to define one or more algebraic operations on this universe. These
operations are shown to satisfy several algebraic properties that may lead
to a characterization of the resulting algebraic structure. These algebraic
structures are then used to assign coordinates to the points of the original
abstract geometry. One of the benefits of this process of coordinatization
is that, having the algebraic structure of the coordinates at hand, enables
one to manipulate the points and lines of the geometry and study several
of their properties by algebraic means, using the operations of the resulting
algebras. For instance, one may define slopes, equations of lines, discover
intersection points by solving systems of equations etc.
Taking the opposite point of view, another significant advantage of this
process is that, given an abstract algebra of the kind used in the coordinatization, there may be a reverse construction for obtaining an abstract
geometry by defining points and lines using the elements of the universe of
the algebraic structure. Then, by studying the structure of the resulting
geometry, and taking into account the interplay between algebra and geometry, one may develop useful geometric intuitions concerning aspects of the
original algebraic structures. So, as it turns out, on several occasions, coor-
COORDINATIZATION IS ALGEBRAIZATION
127
dinatization has positive consequences not only in the study of the original
abstract geometry, by making it amenable to algebraic techniques, but also
in the study of a class of abstract algebras, by possibly endowing their operations with some geometric dimension that may make their presentation
more intuitive and, thus, better understood and more easily and efficiently
studied.
The initial motivation for the attempt to relate the coordinatization
of geometry to the algebraization of logical systems arose from the fact
that the two processes are very similar in intent and nature. As with the
coordinatization of geometry, the goal of abstract algebraic logic, as crystalized in the seminal memoirs monograph “Algebraizable Logics” of Blok
and Pigozzi [2], is to associate with a logical system a class of abstract
algebras in such a way that the logical entailment of the system may be
studied by taking advantage of the algebraic properties of the class of algebras. More precisely, the logical entailment is translated to the equational
entailment of the corresponding class of algebras and, then, its study may
be carried out by a detailed analysis of the structure of the corresponding
algebraic congruences. This can be successfully carried out under the condition that a close enough relationship exists between the original logical
and the induced algebraic entailment systems. Thus, leaving the apparent
discrepancies in the details involved aside, there are essential similarities in
the intention and the goals of coordinatization in geometry and algebraization in logic: both strive to associate with their objects of study (geometries
and logics, respectively) an algebraic system or a class of algebraic systems,
in such a way that the basic structures under consideration (incidence and
consequence, respectively) are reflected in, or can be expressed in terms
of, the algebraic properties of the associated algebraic systems. Then, by
looking at the algebras (algebraic operations and equational consequence,
respectively) one can draw conclusions about the original geometry or logic.
The passage to abstract algebra in both cases is assumed to facilitate the
study of the original objects, since the structure of algebraic objects is, in
many instances, better understood and more readily analyzable than that
of the original objects.
In this paper, we carry out this project of relating the coordinatization
of abstract (affine plane) geometry, as presented in [3], with the process of
algebraization of logics expressed as institutions, as it was developed in the
original papers on this subject [15, 16]. The subfield of categorical abstract
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GEORGE VOUTSADAKIS
algebraic logic (CAAL), whose definitions and techniques we employ in this
work, is the one responsible for making the original concepts, methods and
techniques employed in abstract algebraic logic (see, e.g., [9, 7, 10]) to logics expressed as institutions [11, 12] or π-institutions [8]. In the sense of
CAAL, the algebraization of an institution [16] takes the form of a deductive
equivalence [15] of the institution with another institution whose classes of
models consist of algebraic systems and whose sentences consist of equations. Thus, the way coordinatization is related to algebraization consists,
roughly speaking, of expressing the abstract geometry as a logical system
in the form of an institution, and, then, using the coordinatizing abstract
algebraic structures as algebraic models of another algebraic institution. A
critical selection of sentences of the geometric institution and of equations
of the algebraic institution must be made, so that the coordinatization process may be accurately captured by the process of establishing a deductive
equivalence between the geometric institution and the corresponding algebraic institution. This process involves the construction of translations
between incidence relations and equations and vice-versa, which become
interpretations between entailments on incidence relations and equational
entailments, that are inverses of one another in the precise sense stipulated
by CAAL (see [2] and [15]).
The link established in this way between the coordinatization of abstract
geometry and the general process of algebraization of institutional logics is
interesting for several reasons. First, it reveals a link between two seemingly
unrelated processes: that of associating a class of algebras as an algebraic
semantics to a logical system and that of associating a class of algebras
as coordinate algebras to the models of an abstract geometry. Second, it
sheds new light to the nature of coordinatization by linking it, for instance,
to the association of Boolean algebras to classical propositional calculus
through this very abstract channel. Finally, it enriches abstract algebraic
logic by bringing under its umbrella a very well-known subfield of geometric
investigations, not known previously to be amenable to its methods and
techniques. Thus, the main formal result, Theorem 16, of our work, that
spans both geometry and logic, justifies (at least to some extent) the motto:
“The coordinatization of abstract geometry is a form of algebraization, in the formal sense attributed to this term by the
modern theory of abstract algebraic logic.”
COORDINATIZATION IS ALGEBRAIZATION
129
The paper is organized as follows: In Section 2, abstract geometries, the
geometries of affine planes, are introduced and morphisms between them,
termed geometric morphisms, are defined. In Section 3, an account of the
process of the coordinatization of the affine plane, as detailed in, e.g., [3],
is presented.1 Moreover, coordinate systems are added to abstract geometries and the resulting coordinated abstract geometries and coordinated
geometric morphisms between them are formally defined. The resulting
category is denoted by AG. In Section 4 our main work starts in earnest.
An institution AG is built that formalizes the system of abstract (affine
plane) geometry as a logical system. This involves the construction of its
syntax, consisting of formulas made up using the incidence relation symbol
of the geometry, and of its semantics, which consists of the models of the
axioms of affine plane geometry. The two interact through the satisfaction
relation, which, as is expected, involves an incidence formula being true if,
intuitively speaking, the corresponding intended incidence relation holds in
the chosen geometric model. In Section 5, the properties of the coordinatizing rings of the abstract geometries proven in [3] are revisited together
with the process by which, given such a ring, called a Hall ternary ring, an
abstract geometry may be constructed. If that geometry is endowed with
a canonically associated coordinate system, then the resulting coordinate
ring coincides with the originally given Hall ternary ring. In Section 6,
the institution GA of geometric algebra is constructed. It is essentially the
institution of the equational theory of Hall ternary rings. The fact that
Hall ternary rings provide the coordinate rings for abstract geometry is
the connecting link between this institution and that of abstract geometry,
introduced in Section 4. Indeed, in Section 7, it is shown that GA is an algebraic semantics for AG by exhibiting an interpretation from the sentences
of AG into the equations of GA. Finally, in Section 8, it is shown that an
inverse interpretation from the equations of GA into the sentences of AG
exists. The existence of these two mutually inverse interpretations establish the fact that AG is algebraizable in the sense of CAAL with GA as its
equivalent algebraic semantics. Since AG is the institution of affine plane
geomeries whereas its equivalent algebraic semantics GA is the institution
1
A referee has pointed out that the technical details of the treatment of the algebraization of abstract geometry might become less tedious if a Tarski-Szmielew style
presentation of affine plane geometry [14], using points as the only primitive objects, is
chosen in place of the presentation used in [3], employing both points and lines.
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GEORGE VOUTSADAKIS
of the coordinate rings of affine plane geometries, the deductive equivalence
established between these two institutions reveals the close relationship between coordinatization in geometry and algebraization in abstract algebraic
logic that was displayed in the boxed motto above.
For all unexplained categorical terminology and notation the reader is
encouraged to consult any of the standard references [1, 4, 13].
2. Abstract Geometry and Geometric Morphisms
The notion of abstract geometry (see, e.g., Chapter IV of [3], where this
affine plane geometry is defined) abstracts the very basic features common
in many applied plane geometries, in particular those of many finite geometries and of the Euclidean plane geometry. An abstract geometry
G = hP, L, Ii consists of a set P of abstract points, a set L of abstract
lines, and an incidence relation I ⊆ P × L, which are subject to three
postulates (the term parallel is used to characterize two lines l, l′ ∈ L,
when there does not exist any point p ∈ P incident to both):
Postulate 1 If p0 , p1 ∈ P with p0 6= p1 , then there exists a unique line
l ∈ L incident to both p0 and p1 ;
Postulate 2 If p ∈ P and l ∈ L, such that p is not incident to l, then
there exists a unique line l′ parallel to l, such that p is incident to l′ ;
Postulate 3 There exists at least one quadruple of distinct points in
P , no three of which are incident to the same line.
Given two abstract geometries G = hP, L, Ii and G′ = hP ′ , L′ , I ′ i, a
morphism of abstract geometries (or a geometric morphism, for
short) f : G → G′ is a pair f = hf0 , f1 i, such that f0 : P → P ′ and
f1 : L → L′ , preserving the incidence relations, i.e., satisfying:
pI l
implies
f0 (p) I ′ f1 (l).
A geometric morphism f : G → G′ is said to be strict if the implication
above is an equivalence.
It is shown in Theorem IV.1.2 of [3] that, given an abstract geometry
G = hP, L, Ii, every line l ∈ L of G is incident to the same number2 n of
2
possibly an infinite cardinal
COORDINATIZATION IS ALGEBRAIZATION
131
points and, also, in Corollary IV.1.1 of [3], that every point p ∈ P of G is
incident to the same number n + 1 of lines, i.e., one more than the number
of points incident with any line of G. Moreover, the parallel class of any
one line l ∈ L of G (i.e., the set containing l together with all lines in L
parallel to l) has exactly n members, as many as the number of points on
any one line of the abstract geometry.
3. Coordinatization of an Abstract Geometry
Given an abstract geometry G = hP, L, Ii, there is a well-known process
that can be used to coordinatize its points. We briefly recall this process
here, but the reader is referred to Chapter IV of [3] for many more details.
By Postulate 3 of an abstract geometry, there exists at least one quadruple of distinct points, no three of which are incident with the same line. We
select such a quadruple of points (O, I, X, Y ). The point O is referred to
as the origin, I as the unit point, the unique line incident to O and X is
called the x-line, the unique line incident to O and Y is called the y-line
and the unique line incident to O and I is called the unit line. Next, an
arbitrary set of elements R with the same cardinality n as that of the set of
points on the unit line is picked and a one-to-one correspondence ρ between
the points on the unit line and the elements of R is established. Labels are
assigned to the points of R in such a way that two labels 0 and 1 are reserved for the images of the origin O and the unit point I, respectively, i.e.,
ρ(O) = 0 and ρ(I) = 1. Once this selection has been fixed, a point p ∈ P
can be assigned coordinates c(p) = (a, b) ∈ R2 in the following way:
• If p is incident to the unit line, then c(p) = (ρ(p), ρ(p)). Thus, for
instance, c(O) = (0, 0) and c(I) = (1, 1).
• If p is not incident to the unit line, then, if the unique line incident to
p and parallel to the y-line intersects the unit line at u, with c(u) =
(a, a), and the unique line incident to p and parallel to the x-line
intersects the unit line at v, with c(v) = (b, b), we set c(p) = (a, b).
Furthermore, given a pair (a, b) ∈ R2 , it can be shown that there exists a
unique point p ∈ P of G, such that c(p) = (a, b). We denote that point by
p(a, b). Finally, given m, b ∈ R, it can be shown that there exists a unique
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GEORGE VOUTSADAKIS
line l ∈ L, with slope m and y-intercept b in a way very similar to the
ordinary process used in the Euclidean plane. That line will be denoted by
l(m, b). And, conversely, given a line l, not belonging to the same parallel
class as the y-line, there exist unique m, b ∈ R, such that l has slope m and
y-intercept b. We write d(l) = (m, b).
Given an abstract geometry G = hP, L, Ii, we write S = h(O, I, X, Y ),
R, ρi to denote the coordinate system of G, including the coordinate
set R and the bijection ρ from the points incident to the unit line to R,
described in this section. The pair hG, Si is called a coordinated abstract
geometry.
Let hG, Si and hG′ , S ′ i be two coordinated abstract geometries. A
morphism of coordinated abstract geometries (or, more simply a
coordinated geometric morphism) f¯ : hG, Si → hG′ , S ′ i is a pair
f¯ = hf, f ∗ i, where f : G → G′ is a geometric morphism and f ∗ : R → R′
is a mapping between the corresponding coordinate sets, such that
1. f0 maps O, I, X, Y to O ′ , I ′ , X ′ , Y ′ , respectively;
2. f ∗ (ρ(p)) = ρ′ (f0 (p)), for all p incident to the unit line of hG, Si.
The morphism f¯ is called strict if f : G → G′ is strict.
Proposition 1. Let hG, Si and hG′ , S ′ i be two coordinated abstract
geometries and f¯ : hG, Si → hG′ , S ′ i be a strict coordinated geometric
morphism. Then
1. c′ (f0 (p)) = (f ∗ )2 (c(p)), for all p ∈ P , where (f ∗ )2 denotes coordinatewise application of f ∗ on an order pair;
2. d′ (f1 (l)) = (f ∗ )2 (d(l)), for all l ∈ L, not in the parallel class of the
y-line.
Proof. Suppose that c(p) = (a, b). Thus, the unique line through p
that is parallel to OY intersects OI at p(a, a) and the unique line through
p that is parallel to OX intersects OI at p(b, b). By the definition and the
strictness of f¯, we get that p′ (f ∗ (a), f ∗ (a)) is the point of intersection of
the line through f0 (p) that is parallel to the line O′ Y ′ and, similarly, that
p′ (f ∗ (b), f ∗ (b)) is the point of intersection of the line through f0 (p) that is
parallel to O ′ X ′ . Therefore, we obtain that c′ (f0 (p)) = (f ∗ (a), f ∗ (b)), as
was to be shown. The second part may be proven similarly.
COORDINATIZATION IS ALGEBRAIZATION
133
Proposition 1 shows that the entire strict coordinated geometric morphism f¯ can be reconstructed from knowledge of f ∗ and the pairs hG, Si
and hG′ , S ′ i alone. Thus, in the sequel we will identify a strict coordinated
geometric morphism f¯ = hf, f ∗ i with its second component f ∗ and express
this by saying that f : hG, Si → hG′ , S ′ i is a strict coordinated geometric morphism, where f : R → R′ (i.e., we identify f¯ with f ∗ : R → R′ ,
but, then drop the ∗ from the notation). Moreover, since in what follows
only strict coordinated geometric morphisms will enter our discussion, we
will drop the qualifiers “strict coordinated” and use geometric morphism
to refer to strict coordinated geometric morphisms.
Let AG denote the category with objects all coordinated abstract geometries and morphisms all (strict coordinated) geometric morphisms between them.
For the formulation of the institution of abstract geometry, which will
constitute the cornerstone in the algebraization process that will be presented in the last section of the paper, we will need the following proposition
(for the proof see Sections IV.5 and IV.6 of [3]):
Proposition 2. Let hG, Si be a coordinated abstract geometry. Then,
for all x, m, b ∈ R, there exists a unique y ∈ R, such that p(x, y) I l(m, b).
Following Section IV.5 of [3], we introduce the notation I(x, m, b) to
denote the unique y, such that p(x, y) I l(m, b), whose existence is postulated in the conclusion of Proposition 2. Note that for this notation to be
meaningful, one must have available not only the abstract geometry G =
hP, L, Ii, but also a chosen fixed coordinate system S = h(O, I, X, Y ), R, ρi
on G, even though this is not made explicit in the notation.
4. The Institution AG of Abstract Geometry
In this section, we construct an institution that formalizes the logical system
of abstract (affine plane) geometry. In this institution, the sentences are
constructed using the incidence of the abstract geometry and the models are
coordinated affine planes, accompanied by intended interpretations of the
signature variables in the coordinate spaces of the geometries. Satisfaction
of an incidence formula by a model is defined based on whether the intended
incidence relation denoted by the formula holds in the corresponding model
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GEORGE VOUTSADAKIS
under the stipulated interpretation of its variables in the coordinate space,
which are, subsequently, translated to points and lines of the geometric
model.
Let X be a set. The set of terms Tm(X) over X is recursively defined
as the smallest set, such that
• X ⊆ Tm(X), and
• T (t0 , s0 , s1 ) ∈ Tm(X), for all t0 , s0 , s1 ∈ Tm(X).
The set of formulas Fm(X) over X, on the other hand, is the set
Fm(X) = {(t0 , t1 ) I (s0 , s1 ) : t0 , t1 , s0 , s1 ∈ Tm(X)}.
A function f : X → Tm(Y ) can be uniquely extended to f ∗ : Tm(X) →
Tm(Y ) by setting
• f ∗ (x) = f (x), for all x ∈ X, and
• f ∗ (T (t0 , s0 , s1 )) = T (f ∗ (t0 ), f ∗ (s0 ), f ∗ (s1 )).
Moreover, given such an f : X → Tm(Y ), we define f ∗ : Fm(X) → Fm(Y )
by
f ∗ ((t0 , t1 ) I (s0 , s1 )) = (f ∗ (t0 ), f ∗ (t1 )) I (f ∗ (s0 ), f ∗ (s1 )),
for all t0 , t1 , s0 , s1 ∈ Tm(X).
Define Sign, the signature category of the institution under construction, as the category with objects all small sets and mappings f ∈ Sign(X, Y )
all set mappings f : X → Tm(Y ). Composition in this category is defined
by setting, given f ∈ Sign(X, Y ) and g ∈ Sign(Y, Z),
g ◦ f = g ∗ f : X → Tm(Z).
Let SEN : Set → Set be the functor that maps a set X to the set of
formulas Fm(X) and maps a function f : X → Tm(Y ) to its extension
SEN(f ) = f ∗ : Fm(X) → Fm(Y ).
Construct, next, the functor MOD : Sign → Catop as follows: Given
X ∈ |Sign|, the category MOD(X) has objects all pairs of the form hhG,
Si, σi, where hG, Si is a coordinated abstract geometry, i.e., G = hP, L, Ii
is an abstract geometry and S = h(O, I, X, Y ), R, ρi is a coordinate system
on G, and σ : X → R is a mapping from X to the underlying coordinate
set R of S. Notice that such a mapping σ : X → R extends in a unique
way to a mapping σ ∗ : Tm(X) → R as follows:
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COORDINATIZATION IS ALGEBRAIZATION
• σ ∗ (x) = σ(x), for all x ∈ X, and
• σ ∗ (T (t0 , s0 , s1 )) = I(σ ∗ (t0 ), σ ∗ (s0 ), σ ∗ (s1 )), for all t0 , s0 , s1 ∈ Tm(X).
The morphisms h : hhG, Si, σi → hhG′ , S ′ i, σ ′ i of MOD(X) are (strict
coordinated) geometric morphisms h : hG, Si → hG′ , S ′ i, that make the
following diagram commute:
X
σ
R
h
@
@ σ′
@
R
@
- R′
Given a mapping f : X → Y in Sign (i.e., f : X → Tm(Y ) in Set), the
functor MOD(f ) : MOD(Y ) → MOD(X) is defined, for all hhG, Si, σi ∈
|MOD(Y )|, by
MOD(f )(hhG, Si, σi) = hhG, Si, σ ∗ f i,
and MOD(f )(h) = h : h(G, S), σ ∗ f i → h(G′ , S ′ ), σ ′∗ f i, for all h : hhG, Si,
σi → hhG′ , S ′ i, σ ′ i in MOD(Y ). It is clear that this definition is sound,
since the commutativity of the triangle displayed above implies the commutativity of
X
σ∗f
R
@
@ σ ′∗ f
@
R
@
- R′
h
Finally, define, for all X ∈ |Set|, the relation |=X ⊆ |MOD(X)| × SEN(X),
by setting, for all hhG, Si, σi ∈ |MOD(X)|, with G = hP, L, Ii, and all
(t0 , t1 ) I (s0 , s1 ) ∈ SEN(X),
hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff
σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 )).
Let AG = hSign, SEN, MOD, |=i. AG is called the institution of abstract geometry. This terminology is justified by the following
Theorem 3. The quadruple AG = hSign, SEN, MOD, |=i is an institution.
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GEORGE VOUTSADAKIS
Proof. Suppose f : X → Y be in Sign, (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X)
and hhG, Si, σi ∈ |MOD(Y )|, with G = hP, L, Ii. Then
hhG, Si, σi |=Y
iff
iff
iff
iff
iff
iff
SEN(f )((t0 , t1 ) I (s0 , s1 ))
hhG, Si, σi |=Y (f ∗ (t0 ), f ∗ (t1 )) I (f ∗ (s0 ), f ∗ (s1 ))
σ ∗ (f ∗ (t1 )) = σ ∗ (T (f ∗ (t0 ), f ∗ (s0 ), f ∗ (s1 )))
σ ∗ (f ∗ (t1 )) = I(σ ∗ (f ∗ (t0 )), σ ∗ (f ∗ (s0 )), σ ∗ (f ∗ (s1 )))
(σ ∗ f )∗ (t1 ) = I((σ ∗ f )∗ (t0 ), (σ ∗ f )∗ (s0 ), (σ ∗ f )∗ (s1 ))
hhG, Si, σ ∗ f i |=X (t0 , t1 ) I (s0 , s1 )
MOD(f )(hhG, Si, σi) |=X (t0 , t1 ) I (s0 , s1 ).
Thus, the satisfaction condition holds and AG is an institution.
5. The Ternary Ring R = hR, T, 0, 1i
In this section, we review the abstract properties of the ternary ring R
that is formed by the coordinatization of an abstract geometry G, as presented in Section IV.6 of [3]. Moreover, we describe the reverse process
of coordinatization, presented in Section IV.7 of [3], by which an abstract
geometry G is associated with a ternary ring. The abstract properties of
the coordinatizing ternary rings as well as this construction of an abstract
geometry using the elements and the algebraic properties of a ternary ring
will prove useful when we study the process of algebraization of abstract
geometry in the final sections of this paper.
Let G = hP, L, Ii be an abstract geometry and let S = h(O, I, X, Y ),
R, ρi be a coordinate system for G. Let T : R3 → R be the ternary
operation on R defined by setting T (x, m, b) = I(x, m, b), for all x, m, b ∈ R.
Then, as is shown in Section IV.6 of [3], the operation T on R satisfies the
following properties:
1. T (0, m, b) = T (x, 0, b) = b, for all x, b, m ∈ R.
2. T (x, 1, 0) = T (1, x, 0) = x, for all x ∈ R.
3. The equation T (x, m, b) = T (x, m′ , b′ ) has a unique solution in R, for
all m, m′ , b, b′ ∈ R, with m 6= m′ .
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COORDINATIZATION IS ALGEBRAIZATION
4. The system of equations
(
T (a, x, y) = b
T (a′ , x, y) = b′
)
has a unique solution in R, for all a.a′ , b, b′ ∈ R, with a 6= a′ .
5. The equation T (a, m, x) = c has a unique solution in R, for all
a, m, c ∈ R.
The ternary ring associated with G and the coordinatization S of G will
be denoted by R(G, S) or RS (G). More generally, an algebraic structure
R = hR, T, 0, 1i, where T is a ternary operation satisfying properties 1-5
above, will be called a Hall ternary ring.
Suppose, next, that a Hall ternary ring R is given. Then an abstract
geometry G = hP, L, Ii may be constructed as follows:
• P = {(x, y) : x, y ∈ R};
• L = {{(a, y) : y ∈ R} : a ∈ R}∪{{(x, T (x, m, b)) : x ∈ R} : m, b ∈ R};
• The incidence relation is simply the membership relation, i.e., for all
(x, y) ∈ P and all l ∈ L, we have (x, y) I l iff (x, y) ∈ l.
The following may now be established:
Theorem 4. Given a Hall ternary ring, the structure G is an abstract
geometry. Moreover, the Hall ternary ring associated with G under the
coordinatization S = h((0, 0), (1, 1), (1, 0), (0, 1)), R, iR i coincides with the
Hall ternary ring R.
The coordinated abstract geometry that is obtained in this fashion out
of the given Hall ternary ring R will be denoted by G(R). According to
Theorem 4, we have that R(G(R)) = R.
6. The Institution GA of Geometric Algebra
In this section, the algebraic institution GA, that will serve as the algebraic
semantics of the institution AG of abstract geometry, will be constructed.
Its models will be essentially Hall ternary rings and its sentences will be
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GEORGE VOUTSADAKIS
equations of terms over the language of Hall ternary rings. The fact that
Hall ternary rings serve as coordinate rings for affine plane geometries is
used in subsequent sections to construct mutually inverse interpretations
between these two institutions.
Let Sign be the category of signatures of the institution AG, defined in
Section 4.
Define the functor EQ : Sign → Set as follows: Given a set X,
EQ(X) = {t0 ≈ t1 : t0 , t1 ∈ Tm(X)}
and, given f : X → Tm(Y ) in Sign, define
EQ(f )(t0 ≈ t1 ) = f ∗ (t0 ) ≈ f ∗ (t1 ), for all t0 , t1 ∈ Tm(X),
where f ∗ : Tm(X) → Tm(Y ) is the unique extension of f on terms that
was also defined in Section 4.
Furthermore, define the functor ALG : Sign → Catop as follows: Given
a set X, the category ALG(X) has as its objects all pairs hR, σi, where R
is a Hall ternary ring and σ : X → R an assignment of elements from the
universe R of R to the variables in X. A morphism h : hR, σi → hR′ , σ ′ i
in ALG(X) is a Hall ternary ring homomorphism h : R → R′ , that makes
the following diagram commute:
X
σ
R
@
@ σ′
@
R
@
- R′
h
Moreover, given an f : X → Tm(Y ) in Sign, the corresponding functor
ALG(f ) : ALG(Y ) → ALG(X) sends an object hR, σi ∈ |ALG(Y )| to the
object ALG(f )(hR, σi) = hR, σ ∗ f i and a morphism h : hR, σi → hR′ , σ ′ i
in ALG(Y ) to ALG(f )(h) = h : hR, σ ∗ f i → hR′ , σ ′∗ f i in ALG(X).
Finally, for every set X, define the satisfaction relation
|=X ⊆ |ALG(X)| × EQ(X)
by setting, for all hR, σi ∈ |ALG(X)| and all t0 , t1 ∈ Tm(X),
hR, σi |=X t0 ≈ t1 iff σ ∗ (t0 ) = σ ∗ (t1 ).
The quadruple GA = hSign, EQ, ALG, |=i is called the institution of
geometric algebra, which is justified by the following:
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COORDINATIZATION IS ALGEBRAIZATION
Theorem 5. The quadruple GA = hSign, EQ, ALG, |=i is an institution.
Proof. We check the satisfaction condition. Suppose that f : X →
Tm(Y ) is a morphism in Sign, hR, σi ∈ |ALG(Y )| and t0 ≈ t1 ∈ EQ(X).
Then, we have
hR, σi |=Y EQ(f )(t0 ≈ t1 ) iff
iff
iff
iff
iff
hR, σi |=Y f ∗ (t0 ) ≈ f ∗ (t1 )
σ ∗ (f ∗ (t0 )) = σ ∗ (f ∗ (t1 ))
(σ ∗ f )∗ (t0 ) = (σ ∗ f )∗ (t1 )
hR, σ ∗ f i |=X t0 ≈ t1
ALG(f )(hR, σi) |=X t0 ≈ t1 .
7. GA is an Algebraic Semantics of AG
In this section, it is shown that the institution GA of the equational logic
of the Hall ternary rings, introduced in Section 6, constitutes an algebraic semantics of the institution AG of abstract geometry, introduced
in Section 4. According to the theory of categorical abstract algebraic
logic, an algebraic institution A = hSign′ , SEN′ , MOD′ , |=A i is an algebraic institution semantics of an institution I = hSign, SEN, MOD, |=I i if
the corresponding π-institution π(I) = hSign, SEN, C I i is interpretable in
π(A) = hSign′ , SEN′ , C A i. Note that C I is defined, for all Σ ∈ |Sign| and
all Φ ∪ {φ} ⊆ SEN(Σ) by φ ∈ CΣI (Φ) iff, for every model M ∈ |MOD(Σ)|,
M |=IΣ Φ implies
M |=IΣ φ.
A similar definition applies for C A , i.e., both π(I) and π(A) are the πinstitutions whose closure systems are the closure systems induced by the
semantical entailment systems of the corresponding institutions. The πinstitution π(I) is interpretable in π(A) if there exists an interpretation
hF, αi : I → A, i.e., a functor F : Sign → Sign′ and a natural transformation α : SEN → PSEN′ ◦ F , such that, for all Σ ∈ |Sign| and all
Φ ∪ {φ} ⊆ SEN(Σ),
φ ∈ CΣI (φ)
iff
αΣ (Φ) ⊆ CFA(Σ) (αΣ (φ)).
140
GEORGE VOUTSADAKIS
To show that GA is an algebraic semantics of AG, we define the pair
hISign , αi : AG → GA as follows: ISign : Sign → Sign is the identity
functor on the common signature category of the two institutions. The
natural transformation α : SEN → PEQ is defined by setting, for every set
X and all t0 , t1 , s0 , s1 ∈ Tm(X),
αΣ ((t0 , t1 ) I (s0 , s1 )) = {t1 ≈ T (t0 , s0 , s1 )}.
Lemmas 6 and 7, that follow, are supporting lemmas for showing the
main equivalence establishing the interpretation property of hISign , αi. This
equivalence is shown in Proposition 8. The main theorem, Theorem 9,
simply restates the equivalence of Proposition 8 in the language of abstract
algebraic logic.
Lemma 6. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 , s0 , s1 ∈
Tm(X). Then
hR, σi |=X t1 ≈ T (t0 , s0 , s1 )
iff
hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ).
Proof.
hR, σi |=X t1 ≈ T (t0 , s0 , s1 )
iff
iff
iff
iff
σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 ))
σ ∗ (t1 ) = I(σ ∗ (t0 ), σ ∗ (s0 ), σ ∗ (s1 ))
(σ ∗ (t0 ), σ ∗ (t0 )) I (σ ∗ (s0 ), σ ∗ (s1 ))
hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ).
Lemma 7. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and t0 , t1 , s0 , s1 ∈
Tm(X). Then
hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 )
iff
hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ).
Proof.
hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 )
iff σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 ))
iff hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ).
COORDINATIZATION IS ALGEBRAIZATION
141
Proposition 8. Let X be a set. For all Φ ∪ {φ} ⊆ SEN(X),
AG
φ ∈ CX
(Φ)
iff
GA
αX (φ) ⊆ CX
(αX (Φ)).
AG
(Φ). This means that, for all
Proof. Assume, first, that φ ∈ CX
hhG, Si, σi ∈ |MOD(X)|, hhG, Si, σi |=X Φ implies hhG, Si, σi |=X φ.
Suppose, now, that hR, σi ∈ |ALG(X)|, and hR, σi |=X αX (Φ). Thus,
hR, σi |=X t1 ≈ T (t0 , s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈ Φ. Therefore, by
Lemma 6, we have hG(R), σi |=X (t0 , t1 ) I (s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈
Φ. Hence, by hypothesis, we get that hG(R), σi |=X φ. Again, using
Lemma 6, we obtain that hR, σi |=X αX (φ). This proves that αX (φ) ⊆
GA
(αΣ (Φ)).
CX
GA
(αX (Φ)). This means that,
Assume, conversely, that αX (φ) ⊆ CX
for all hR, σi ∈ |ALG(X)|, hR, σi |=X αX (Φ) implies hR, σi |=X αX (φ).
Suppose, now, that hhG, Si, σi ∈ |MOD(X)|, such that hhG, Si, σi |=X
Φ. Thus, hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ), for all (t0 , t1 ) I (s0 , s1 ) ∈
Φ. Therefore, by Lemma 7, hRS (G), σi |=X t1 ≈ T (t0 , s0 , s1 ), for all
(t0 , t1 ) I (s0 , s1 ) ∈ Φ. Hence, by hypothesis, we get that hRS (G), σi |=X
αX (φ). Again, using Lemma 7, we obtain that hhG, Si, σi |=X φ. This
AG
(Φ).
proves that φ ⊆ CX
Theorem 9. The institution of Hall ternary rings GA is an algebraic
semantics of the institution AG of abstract geometry.
Proof. This is just a restatement of Proposition 8.
8. Algebraization of AG
In this section, it is shown that not only is GA an algebraic semantics of
AG, but, moreover, AG is an algebraizable institution with GA its algebraic counterpart. We do this by showing that there exists an interpretation hISign , βi from GA into AG, which is inverse to the interpretation
hISign , αi. Thus, the two institutions AG and GA are deductively equivalent
institutions, as is required for algebraizability.
To this end, define the natural transformation β : EQ → PSEN, by
setting, for every set X and all terms t0 , t1 ∈ Tm(X),
βX (t0 ≈ t1 ) = {(t0 , t1 ) I (1, 0)}.
142
GEORGE VOUTSADAKIS
Then we have the following analogs of Lemmas 6 and 7 and of Proposition
8 establishing that hISign , βi is an interpretation from GA to AG:
Lemma 10. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 Tm(X).
Then
hR, σi |=X t0 ≈ t1 iff hG(R), σi |=X (t0 , t1 ) I (1, 0).
Proof.
hR, σi |=X t0 ≈ t1 iff
iff
iff
iff
σ ∗ (t0 ) = σ ∗ (t1 )
σ ∗ (t0 ) = I(σ ∗ (t1 ), 1, 0)
σ ∗ (t0 ) = σ ∗ (T (t1 , 1, 0))
hG(R), σi |=X (t0 , t1 ) I (1, 0).
Lemma 11. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and t0 , t1 ∈
Tm(X). Then
hhG, Si, σi |=X (t0 , t1 ) I (1, 0)
iff
hRS (G), σi |=X t0 ≈ t1 .
hhG, Si, σi |=X (t0 , t1 ) I (1, 0) iff
iff
iff
iff
σ ∗ (t0 ) = σ ∗ (T (t1 , 1, 0))
σ ∗ (t0 ) = I(σ ∗ (t1 ), 1, 0)
σ ∗ (t0 ) = σ ∗ (t1 )
hRS (G), σi |=X t0 ≈ t1 .
Proof.
Proposition 12. Let X be a set. For all E ∪ {t0 ≈ t1 } ⊆ EQ(X),
GA
t 0 ≈ t 1 ∈ CX
(E)
iff
AG
βX (t0 ≈ t1 ) ⊆ CX
(βX (E)).
GA
(E). This means that, for
Proof. Assume, first, that t0 ≈ t1 ∈ CX
all hR, σi ∈ |ALG(X)|, hR, σi |=X E implies hR, σi |=X t0 ≈ t1 . Suppose,
now, that hhG, Si, σi ∈ |MOD(X)|, and hhG, Si, σi |=X βX (E). Thus,
hhG, Si, σi |=X (ǫ0 , ǫ1 ) I (1, 0), for all ǫ0 ≈ ǫ1 ∈ E. Therefore, by Lemma
11, hRS (G), σi |=X ǫ0 ≈ ǫ1 , for all ǫ0 ≈ ǫ1 ∈ E. Hence, by hypothesis,
we have hRS (G), σi |=X t0 ≈ t1 . Again, using Lemma 11, we obtain that
AG
(βΣ (E)).
hhG, Si, σi |=X βX (t0 ≈ t1 ). This proves that βX (t0 ≈ t1 ) ⊆ CX
AG
Assume, conversely, that βX (t0 ≈ T1 ) ⊆ CX (βX (E)). This means
that, for all hhG, Si, σi ∈ |MOD(X)|, hhG, Si, σi |=X βX (E) implies hhG, Si,
COORDINATIZATION IS ALGEBRAIZATION
143
σi |=X βX (t0 ≈ t1 ). Suppose, now, that hR, σi ∈ |ALG(X)|, such that
hR, σi |=X E. Thus, hR, σi |=X ǫ0 ≈ ǫ1 , for all ǫ0 , ≈ ǫ1 ∈ E. Therefore, by
Lemma 10, hG(R), σi |=X (ǫ0 , ǫ1 ) I (1, 0), for all ǫ0 ≈ ǫ1 ∈ E. Hence, by
hypothesis, we get that hG(R), σi |=X βX (t0 ≈ t1 ). Again, using Lemma
GA
(E).
10, we obtain that hR, σi |=X t0 ≈ t1 . This proves that t0 ≈ t1 ∈ CX
Theorem 13. The pair hISign , βi forms an interpretation from the institution GA of Hall ternary rings to the institution AG of abstract geometry.
Proof. This is simply a restatement of Proposition 12.
To complete our demonstration that AG is algebraizable and the institution of Hall ternary rings GA is its equivalent algebraic semantics, it
suffices now to show that the two interpretations hISign , αi : AG → GA and
hISign , βi : GA → AG are inverse of one another in the precise technical
sense of [15] (see also [2]). In other words, it must be shown that composing α and β results in interderivable sets of geometric formulas in AG and
composing β and α results in interderivable sets of equations in GA. The
following two lemmas pave the way for the final results:
Lemma 14. Let X be a set, hhG, Si, σi ∈ |MOD(X)| and
(t0 , t1 ) I (s0 , s1 ) ∈ SEN(X).
Then
hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 )
iff hhG, Si, σi |=X (t1 , T (t0 , s0 , s1 )) I (1, 0).
Proof.
hhG, Si, σi |=X (t0 , t1 ) I
iff
iff
iff
iff
(s0 , s1 )
σ ∗ (t1 ) = σ ∗ (T (t0 , s0 , s1 ))
σ ∗ (T (t0 , s0 , s1 )) = I(σ ∗ (t1 ), 1, 0)
σ ∗ (T (t0 , s0 , s1 )) = σ ∗ (T (t1 , 1, 0))
hhG, Si, σi |=X (t1 , T (t0 , s0 , s1 )) I (1, 0).
Lemma 15. Let X be a set, hR, σi ∈ |ALG(X)| and t0 , t1 ∈ Tm(X).
Then
hR, σi |=X t0 ≈ t1 iff hR, σi |=X t1 ≈ T (t0 , 1, 0).
144
GEORGE VOUTSADAKIS
Proof.
hR, σi |=X t0 ≈ t1 iff
iff
iff
iff
σ ∗ (t0 ) = σ ∗ (t1 )
σ ∗ (t1 ) = I(σ ∗ (t0 ), 1, 0)
σ ∗ (t1 ) = σ ∗ (T (t0 , 1, 0))
hR, σi |=X t1 ≈ T (t0 , 1, 0).
Theorem 16. The institution AG of abstract geometry and the institution GA of Hall ternary rings are deductively equivalent institutions.
Thus, AG is algebraizable and the institution GA is its equivalent algebraic
semantics.
Proof. We have already proven in Theorems 9 and 13 that hISign , αi :
AG → GA and hISign , βi : GA → AG are interpretations. Thus, it suffices
to show that they are inverse to one another. Suppose, that X is a set
and (t0 , t1 ) I (s0 , s1 ) ∈ SEN(X). Then, for all hhG, Si, σi ∈ |MOD(X)|, we
have, by Lemma 14, that hhG, Si, σi |=X (t0 , t1 ) I (s0 , s1 ) iff hhG, Si, σi |=X
AG
((t0 , t1 ) I (s0 , s1 )) =
βX (αX ((t0 , t1 ) I (s0 , s1 ))). Therefore, we get that CX
AG
CX (βX (αX ((t0 , t1 ) I (s0 , s1 )))). Similarly, if X is a set and t0 ≈ t1 ∈
EQ(X), then, for all hR, σi ∈ |ALG(X)|, we have, by Lemma 15, that
hR, σi |=X t0 ≈ t1 iff hR, σi |=X αX (βX (t0 ≈ t1 )). Therefore, we get that
GA
GA
(αX (βX (t0 ≈ t1 ))). This concludes the proof that
(t0 ≈ t1 ) = CX
CX
hISign , αi and hISign , βi are indeed inverse to one another in the precise
technical sense of abstract algebraic logic
Theorem 16, which is the main theorem of the paper, shows that the
institution of abstract (affine plane) geometry is deductively equivalent to
the institution of geometric algebra. Thus, the class of all Hall ternary
rings forms an equivalent algebraic semantics of affine plane geometry in
the precise technical sense of abstract algebraic logic. It is in this sense
that one may say that the coordinate rings of modern abstract geometry
form an equivalent algebraic semantics of the logic of abstract geometry.
Therefore, the process of coordinatization in geometry may be viewed as a
special case of the formal process of algebraization of logical systems.
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COORDINATIZATION IS ALGEBRAIZATION
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School of Mathematics and Computer Science,
Lake Superior State University,
Sault Sainte Marie, MI 49783, USA
[email protected]